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equicontinuous family of functions

Equicontinuous functions

Context

Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

Equicontinuous functions

Idea

A function ff is continuous if, roughly speaking, f(x)f(x) is arbitrarily close to f(y)f(y) whenever xx is sufficiently close to yy. However, ‘close’ is relative, and f(x)f(x) may be much closer to f(y)f(y) than g(x)g(x) is to g(y)g(y), even if both ff and gg are continuous. Nevertheless, given a family of functions, we may have that f(x)f(x) is arbitrarily close to f(y)f(y) for every function ff in the family at once whenever xx is sufficiently close to yy. In this case, the family of functions is equicontinuous.

Definitions

Because we are considering the relative degree of closeness between potentially unrelated pairs of points, we need a uniform structure to define this concept. So let XX and YY be uniform spaces (although the concept should make sense in somewhat greater generality), and let \mathcal{F} be a family of functions from XX to YY.

Definition

The family \mathcal{F} is continuous if each member is continuous: For each entourage EE in YY, for each function ff in \mathcal{F} and each point xXx \in X, for some entourage DD in XX, for each point yy in XX, whenever x Dyx \approx_D y, we have f(x) Ef(y)f(x) \approx_E f(y).

In short:

E,f,x,D,y,x Dyf(x) Ef(y). \forall E,\; \forall f,\; \forall x,\; \exists D,\; \forall y,\; x \approx_D y \;\Rightarrow\; f(x) \approx_E f(y) .
Definition

The family \mathcal{F} is uniformly continuous if each member is uniformly continuous: For each entourage EE in YY, for each function ff in \mathcal{F}, for some entourage DD in XX, for each point xx in XX, for each point yy in XX, whenever x Dyx \approx_D y, we have f(x) Ef(y)f(x) \approx_E f(y).

In short:

E,f,D,x,y,x Dyf(x) Ef(y). \forall E,\; \forall f,\; \exists D,\; \forall x,\; \forall y,\; x \approx_D y \;\Rightarrow\; f(x) \approx_E f(y) .
Definition

The family \mathcal{F} is equicontinuous if: For each entourage EE in YY, for each point xx in XX, for some entourage DD in XX, for each function ff in \mathcal{F}, for each point yy in XX, whenever x Dyx \approx_D y, we have f(x) Ef(y)f(x) \approx_E f(y).

In short:

E,x,D,f,y,x Dyf(x) Ef(y). \forall E,\; \forall x,\; \exists D,\; \forall f,\; \forall y,\; x \approx_D y \;\Rightarrow\; f(x) \approx_E f(y) .
Definition

The family \mathcal{F} is uniformly equicontinuous if: For each entourage EE in YY, for some entourage DD in XX, for each function ff in \mathcal{F} and each point xx in XX, for each point yy in XX, whenever x Dyx \approx_D y, we have f(x) Ef(y)f(x) \approx_E f(y).

In short:

E,D,f,x,y,x Dyf(x) Ef(y). \forall E,\; \exists D,\; \forall f,\; \forall x,\; \forall y,\; x \approx_D y \;\Rightarrow\; f(x) \approx_E f(y) .

All of these definitions are identical except for the placement of the quantifiers f\forall f and x\forall x before or after the quantifier D\exists D.

Just as it is reasonable to speak of a single function ff continuous at a single point xx, so it is reasonable to speak of a family equicontinuous at a single point xx (or, for that matter, of a single function ff uniformly continuous). However, it makes no sense to speak of a single function ff equicontinuous (or uniformly equicontinuous), nor can we speak of a family of functions uniformly equicontinuous at a single point xx.

Properties

Cauchy sum theorem

The Cauchy sum theorem holds for an equicontinuous family of functions (from the real line to itself), without the requirement for uniform convergence.

Arzela-Ascoli theorem

The importance of equicontinuity is perhaps best illustrated by the Arzelà-Ascoli theorem, which gives conditions for a set in a function space to be compact. A reasonably general version is this:

Theorem

Let XX be a convergence space and YY a uniform space. Then a subset FC(X,Y)F \subseteq C(X, Y) is relatively compact (has compact closure) iff it is equicontinuous and {f(x):fF}\{f(x):\; f \in F\} is relatively compact in YY for each xXx \in X.

See BB, corollary 2.4.9. Here the topology on the space C(X,Y)C(X, Y) of continuous functions XYX \to Y is the so-called natural topology, namely the largest topology on C(X,Y)C(X, Y) such that for all spaces AA, the continuity of a map g:A×XYg: A \times X \to Y implies the continuity of its transpose g^:AC(X,Y)\hat{g}: A \to C(X, Y). This is the same as the exponential in TopTop whenever the exponential exists. See Escardó, sections 8.1 and 10.2; see also ELS where it is shown that the natural topology on C(X,Y)C(X, Y) coincides with the topology of continuous convergence (which is the context for the theorem above).

(Some applications of Arzela-Ascoli should also be given.)

Representability

It happens that the property of a family of functions being (uniformly) equicontinuous is equivalent to them defining a single function which is (uniformly) continuous.

Recall that for a family of functions f i:ABf_i : A \to B for iIi \in I, we can define a function f^:AB I\hat f : A \to B^I by f^(a)(i)=f i(a)\hat f(a)(i) = f_i(a), and that the f if_i are all (uniformly) continuous iff f^\hat f is (uniformly) continuous when B IB^I is given the product uniformity.

However, we can give B IB^I another uniformity more analogous to the box topology of topological spaces. Specifically, the box power B IB^{\square I} is the uniform space with point set B IB^I, and where EB I×B IE \subseteq B^I \times B^I is an entourage iff there is some entourage GG on BB such that for all f Egf \sim_E g in B IB^I and all iIi \in I, f(i) Gg(i)f(i) \sim_G g(i). This means that the entoruages of B IB^{\square I} are generated by G IG^I for entourages GG on BB.

With this definition, we can see that the family of functions f i:ABf_i : A \to B is (uniformly) equicontinuous iff f^:AB I\hat f : A \to B^{\square I} is (uniformly) continuous.

This also gives a convenient way to show that an equicontinuous net of continuous functions which converge pointwise converges to a continuous function: a net of equicontinuous functions f d:ABf_d : A \to B gives a continuous function AB DA \to B^{\square D}. By the below proposition, if the f df_d converge pointwise, they converge to a continuous function.

Proposition

Let BB be a separated uniform space and DD a directed set, let conv(D,B)conv(D,B) be the subspace of B DB^{\square D} which contains the convergent nets. Then the map lim:conv(D,B)Blim : conv(D,B) \to B which takes a net to the point it converges to is continuous.

Proof

Let MM in conv(D,B)conv(D,B) and limMUBlim M \in U \subset B be open. Then there is an entourage EE on BB such that (EE)[limM]U (E \circ E)[lim M] \subset U . E D[M]:={Ni.M(i) EN(i)}conv(D,B)E^D[M] := \{N \mid \forall i. M(i) \sim_E N(i) \} \subset conv(D,B) is a neighbourhood of MM. For any NE D[M]N \in E^D[M], we have that limNE[limM]¯lim N \in \widebar{E[lim M]} (as all N(i)N(i) are in this closed set), so limNE[x]¯\(EE)[x]Ulim N \in \widebar{E[x]} \subseteq \(E \circ E)[x] \subseteq U. This means that limlim is continuous at MM, as required.

References

  • Wikipedia, Equicontinuity

  • R. Beattie and H.-P. Butzmann, Convergence Structures and Applications to Functional Analysis, Kluwer Academic Publishers (2002).

  • Martín Escardó, Synthetic topology of data types and classical spaces, Electronic Notes in Theor. Comp. Sci. (2004). (pdf)
  • Martín Escardó, Jimmie Lawson, and Alex Simpson, Comparing cartesian closed categories of (core) compactly generated spaces, Topology and its Applications Vol. 143 Iss. 1-3 (2004), 105-145. (web)

Last revised on November 17, 2021 at 15:33:14. See the history of this page for a list of all contributions to it.